Just get a shovel and dig the ditch yourself. Great exercise! You may be able to call a "locate service" and they will mark where your sewer is for free. This is provided by the utilities. Gas, electric, water, etc. They don't want anyone digging into their lines, so they are happy to mark where they are before digging for projects.
Anyway it may take a week or a month of digging yourself. Just do what you can each day. But I guarantee you will feel great after all that exercise. It is the same as digging a hole to plant a tree, you just keep going!
Just be sure to cover the hole with boards after done working everyday so no one falls in.
Then replacing the sewer line is not rocket science. It is just a plastic pipe and clamps. Should be plenty on this topic on the internet. They can also give helpful advice at the hardware or home improvement store.
FYI - My dad always told me if I did not do well in school I would be a ditch digger! So I enjoy digging ditches and thinking of him...
If the plumber is supplying the backhoe its not too bad of a price.
Trenchless, as in a liner or pipe burst probably won't help you because your at the 6x4 transition. It would probably cost more anyways. Are you removing sidewalks, driveway or working in the city right of way?